# Properties

 Label 2-546-273.257-c1-0-37 Degree $2$ Conductor $546$ Sign $-0.754 - 0.656i$ Analytic cond. $4.35983$ Root an. cond. $2.08802$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + (−1.5 − 0.866i)3-s + 4-s + (−3 − 1.73i)5-s + (−1.5 − 0.866i)6-s + (−0.5 − 2.59i)7-s + 8-s + (1.5 + 2.59i)9-s + (−3 − 1.73i)10-s + (−3 + 5.19i)11-s + (−1.5 − 0.866i)12-s + (−2.5 + 2.59i)13-s + (−0.5 − 2.59i)14-s + (3 + 5.19i)15-s + 16-s + ⋯
 L(s)  = 1 + 0.707·2-s + (−0.866 − 0.499i)3-s + 0.5·4-s + (−1.34 − 0.774i)5-s + (−0.612 − 0.353i)6-s + (−0.188 − 0.981i)7-s + 0.353·8-s + (0.5 + 0.866i)9-s + (−0.948 − 0.547i)10-s + (−0.904 + 1.56i)11-s + (−0.433 − 0.249i)12-s + (−0.693 + 0.720i)13-s + (−0.133 − 0.694i)14-s + (0.774 + 1.34i)15-s + 0.250·16-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $-0.754 - 0.656i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (257, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$1$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ -0.754 - 0.656i)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - T$$
3 $$1 + (1.5 + 0.866i)T$$
7 $$1 + (0.5 + 2.59i)T$$
13 $$1 + (2.5 - 2.59i)T$$
good5 $$1 + (3 + 1.73i)T + (2.5 + 4.33i)T^{2}$$
11 $$1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2}$$
17 $$1 + 17T^{2}$$
19 $$1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + 3.46iT - 23T^{2}$$
29 $$1 + (6 - 3.46i)T + (14.5 - 25.1i)T^{2}$$
31 $$1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + 8.66iT - 37T^{2}$$
41 $$1 + (9 - 5.19i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (6 + 3.46i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 + (3 - 1.73i)T + (26.5 - 45.8i)T^{2}$$
59 $$1 - 59T^{2}$$
61 $$1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (3 + 1.73i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 3.46iT - 83T^{2}$$
89 $$1 + 3.46iT - 89T^{2}$$
97 $$1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$